Question

The formula for the volume of a right circular cylinder is [tex]V=\pi r^2 h[/tex]. If [tex]r=2b[/tex] and [tex]h=5b+3[/tex], what is the volume of the cylinder in terms of [tex]b[/tex]?

A. [tex]10\pi b^2 + 6\pi b[/tex]
B. [tex]20\pi b^3 + 12\pi b^2[/tex]
C. [tex]20\pi^2 b^3 + 12\pi^2 b^2[/tex]
D. [tex]50\pi b^3 + 20\pi b^2 + 90\pi b[/tex]



Answer :

To find the volume of the cylinder in terms of [tex]\(b\)[/tex], let's proceed step-by-step using the given relations and the formula for the volume of a right circular cylinder.

1. Given Relations:
- [tex]\( r = 2b \)[/tex]
- [tex]\( h = 5b + 3 \)[/tex]

2. Volume Formula for a Cylinder:
[tex]\[ V = \pi r^2 h \][/tex]

3. Substitute the given values of [tex]\(r\)[/tex] and [tex]\(h\)[/tex] into the volume formula:
[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]

4. Simplify the expression:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]

5. Distribute [tex]\(4b^2\)[/tex] within the parentheses:
[tex]\[ V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]

6. Perform the multiplication:
[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]

7. Combine terms:
[tex]\[ V = 20 \pi b^3 + 12 \pi b^2 \][/tex]

Hence, the volume of the cylinder in terms of [tex]\(b\)[/tex] is:

[tex]\[ V = 20 \pi b^3 + 12 \pi b^2 \][/tex]

Therefore, the correct answer from the given options is:

[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]